An annular combustion chamber for a turbomachine is disclosed. The chamber includes an inner wall, an outer wall, and a chamber end wall disposed between the inner and outer walls in the upstream region of the chamber. The chamber end wall presents an outer fastener rim and/or an inner fastener rim, and the outer and/or inner wall presents a respective upstream fastener rim. The chamber end wall and the outer and/or inner wall are fastened together via their fastener rims. Cooling channels are made between the fastener rims and open out to the inside of the combustion chamber. Advantageously, a spacer is placed between the fastener rims, and the channels are formed in the thickness or in the sides of the spacer.